The difference between Dyson DC59 and DC62 may be the model number.
If you visit the official websites of Dyson for different countries, you can find that the markets that have the DC59 available do not have the DC62, and the countries where you can find a DC62 do not sell the DC59.
And if you compare the features and specs of these 2 Dyson cordless vacuum cleaners, you will find that they are almost the same unit. For example, they both feature 350W Dyson digital motor V6, 2 Tier Radial cyclones, improved motorized head with carbon fibre filaments, re-engineered nickel manganese cobalt battery to provides more power than previous models, etc.

So, I think they sell the same unit with different model numbers to different markets. For example, at Dyson AU websites, you can find the DC59 www.dyson.com.au/vacuumcleaners/handstick/dc59/dc59.aspx; at Dyson CA you will find the DC62 model http://www.dysoncanada.ca/en-ca/vacuumcleaners/cordless/dc62/dc62-animal.aspx

Do you know more details about them? If yes, please share with us by commenting below.

We will compare Dyson DC59 and DC58 within this post.
Both of them are cordless. Dyson DC59 come with a detachable wand and floor tool that make it can be used as a handheld or stick vacuum cleaner, while the DC58 does not include these tools, it is just a hand vacuum. And the DC59 has a wall mounting bracket while the DC58 doesn’t.

Dyson DC59 VS DC58

	Dyson DC58	Dyson DC59/DC59 Animal
Images
Motor	350W, Dyson digital motor V6	350W, Dyson digital motor V6
Cyclone Technology Type	2 Tier Radial™ cyclone technology	2 Tier Radial™ cyclone technology
Multiple Floor Types Cleaning	NA	Yes
Dual Power Mode	Yes	Yes
Suction Power	28 airwatts at Standard Power or 100 airwatts at BOOST mode	28 airwatts at standard power or 100 airwatts in boost mode
Bin Capacity	0.12 gallons	0.12 gallons
Bag Type	Bagless	Bagless
Battery Type	Nickel Manganese Cobalt/Li-ion	Nickel Manganese Cobalt/Li-ion
Battery Life	20 minutes on standard mode, 6 minutes Boost mode for more difficult tasks.	26 minutes on standard mode, 6 minutes cleaning time on boost mode
Charge Time	3.5 hours	3.5 hours
Cleaner Head	The new motorized cleaner head of DC59 is more effective on picking up dust and dirt across carpeting and hard floors.
Cleaner Head Width	NA	9.8"
Docking Station	No	Yes
Battery Charger	Yes	Yes
Removable Wand	No	Yes
Floor Tool	No	Yes, the floor tool can be used to pick up dust and dirt from carpet and hard floors.
Combination Tool	Yes	Yes
Crevice Tool	Yes	Yes
Warranty	2 year warranty on parts & labor	2 year warranty on parts & labor
Weight	3.4 lbs	4.6 lbs
Dimensions	8.2" x 15.6" x 5.7" (HxLxW)	9.8" x 47.8" x 8.2"
Official Price	$249.99 on Dyson.com on March 30, 2014	Dyson DC59 Animal is priced at $499.99 on Dyson.com on March 30, 2014

In this post, DC59 stands for DC59 Animal Cordless Cleaner while the DC58 stands for Dyson DC58 Handheld Vacuum.

If you know other difference between Dyson DC58 and DC59, please share here.

The Dyson DC59 is better. It uses the new 2 Tier Radial cyclones technology and equips with better motor that makes it generate more powerful suction; improved battery to provides longer battery life; some parts of the unit are redesigned to make it more convenient to use such as the filter design and Trigger design; improved floor tool is more stable, picking up more, with less hair clog problem, etc.

Dyson DC59 VS DC35

	Dyson DC59 Animal	Dyson DC35 Multi Floor
Image
Motor	Dyson digital motor V6, provides greater suction.	Dyson digital motor V2
Dual Power Mode	Yes, the DC59 has 50% more power in boost mode than DC35.	Yes
Suction Power	28 airwatts at standard power or 100 airwatts in boost mode	28 airwatts at end of wand in standard mode or 65 airwatts at end of wand in max mode
Multiple Floor Cleaning	Yes	Yes
Cleaner Head	Wider cleaner head and is more powerful than DC35, 9.8 inch width	8.25 inch wide
Bin Capacity	0.12 gallons	smaller bin, 0.09 gallons
Bag Type	Bagless	Bagless
Battery Life	Battery life of DC59 is 26 minutes on regular speed or 6 minutes on boost.	DC35 has a 15 minute run time or 6 minutes on MAX power.
Battery Type	Re-engineered Nickel Manganese Cobalt/Li-ion transmits 1.5 times more power than DC44.	Lithium-ion battery
Charge Time	3.5 hours	3.5 hours
Docking Station	Yes	Yes
Tools Included	Mini motorized tool, Combination accessory tool, Crevice tool	Includes a crevice tool and a combination accessory tool
Cyclone Technology Type	2 Tier Radial™ cyclones, increase airflow and capture fine dust.	Root Cyclone™ technology
Warranty	2 year parts & labor warranty	2 year parts & labor warranty
Dimensions	47.8" x 8.2" x 9.8"	44.1" x 9.1" x 11.8"
Weight	4.6 pounds	4.85 lbs

If you know other difference between Dyson DC59 and DC35, please share, thank you.

Dyson DC59 VS DC44

	Dyson DC44 Animal	Dyson DC59/DC59 Animal
Images
Motor	Dyson digital motor V2	Dyson digital motor V6, enable the machine to generate more powerful suction.
Cyclone Technology Type	Root Cyclone™ technology	2 Tier Radial™ cyclone technology, increase airflow and capture fine dust.
Multiple Floor Types Cleaning	Yes	Yes
Dual Power Mode	Yes	Yes
Suction Power	28 airwatts in standard mode or 65 airwatts in max power mode	28 airwatts at standard power or 100 airwatts in boost mode
Trigger Design	Trigger of DC59 is redesigned, now it is lighter to squeeze than the DC44.
Bin Capacity	0.09 gallons	0.12 gallons, DC59 equip with larger dust bin.
Bag Type	Bagless	Bagless
Filter Design	The filter of DC59 is easier to pull out and wash.
Battery Type	Nickel Manganese Cobalt/Li-ion	Re-engineered Nickel Manganese Cobalt/Li-ion transmits 1.5 times more power than DC44.
Battery Life	DC44 has a run time of 20 minutes on standard speed or 8 minutes on Max mode.	Battery life of DC59 is 26 minutes on regular speed or 6 minutes on boost.
Charge Time	5.5 hours	3.5 hours
Location of Charging Indicator	Charging indicator of DC44 is located on the power adapter.	Charging indicator light of DC59 is on the unit body.
Cleaner Head	The new motorized cleaner head of DC59 is more effective on picking up dust and dirt across carpeting and hard floors.
Cleaner Head Width	8.25"	9.8"
Docking Station	Yes	Yes
Wall Mountable	Yes	Yes
Warranty	2 year parts & labor warranty	2 year parts & labor warranty
Weight	5.07 lbs	4.6 lbs DC59 is lighter to carry.
Dimensions	44.1" x 9.1" x 11.8"	47.8" x 8.2" x 9.8"

If you know more difference between Dyson DC44 AND DC59, please share below.